Econ Theory (2014) 57:377–405 DOI 10.1007/s00199-014-0826-y RESEARCH ARTICLE

Bivariate almost stochastic dominance Michel M. Denuit · Rachel J. Huang · Larry Y. Tzeng

Received: 17 September 2013 / Accepted: 13 June 2014 / Published online: 26 June 2014 © Springer-Verlag Berlin Heidelberg 2014

Abstract Univariate almost stochastic dominance has been widely studied and applied since its introduction by Leshno and Levy (Manag Sci 48:1074–1085, 2002). This paper extends this construction to the bivariate case by means of suitable twoattribute utility functions. After having confined correlation aversion and correlation loving to some acceptable levels, bivariate almost stochastic dominance rules are introduced for the preferences exhibiting confined correlation aversion and confined correlation loving. The impact of a change in risk in terms of bivariate almost stochastic dominance on optimal saving is analyzed as an application, as well as the effect of envy and altruism on income distributions. Finally, alternative definitions of bivariate almost stochastic dominance are discussed, as well as testing procedures for such dominance rules in financial problems.

The authors would like to express their gratitude to an anonymous referee whose suggestions have been extremely helpful to revise previous versions of the present work. Michel Denuit acknowledges the financial support from the contract “Projet d’Actions de Recherche Concertées” No 12/17-045 of the “Communauté française de Belgique,” granted by the “Académie universitaire Louvain”. M. M. Denuit (B) Institut de Statistique, Biostatistique et Sciences Actuarielles (ISBA), Université Catholique de Louvain, Louvain-la-Neuve, Belgium e-mail: [email protected] R. J. Huang Department of Finance, National Central University, Taoyuan, Taiwan e-mail: [email protected] L. Y. Tzeng Department of Finance, National Taiwan University, Taipei, Taiwan e-mail: [email protected]

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Keywords Almost stochastic dominance · Correlation aversion · Correlation loving · Optimal saving JEL Classification

D81

1 Introduction Many decision problems involve several commodities, often called attributes. The trade-off between them commonly captured by the mixed derivatives of the utility function cannot be ignored in general as it plays an important role and may even reverse choices, as demonstrated by Kaplanski and Levy (2013). The literature has employed multivariate stochastic dominance to deal with the various dimensions of decision making such as multi-period portfolio choices (Levy and Parouch 1974) or social welfare as a function of income and life expectancy or income and quality of life (Atkinson and Bourguignon 1982), for instance. Relevant references include Richard (1975), Epstein and Tanny (1980), Eeckhoudt et al. (2007), Tsetlin and Winkler (2009) and Denuit et al. (2013). Despite their theoretical attractiveness, stochastic dominance rules may create paradoxes in the sense that they fail to distinguish between some risky prospects, whereas it is obvious that the vast majority of agents would prefer one over the other. To overcome this drawback of univariate stochastic dominance, Leshno and Levy (2002) and Tzeng et al. (2013) proposed excluding some extreme preferences in the given class since such preferences presumably rarely represent real-world decision makers’ preferences. They further established a concept of “almost stochastic dominance” for most decision makers to rank distributions. Almost stochastic dominance allows for some violation of stochastic dominance and thus substantially increases the applicability in practice. Bali et al. (2009, 2011, 2013) and Levy (2009) have provided evidence to show that almost stochastic dominance rules can support several popular investment practices which cannot be supported by stochastic dominance rules. The purpose of this paper is to extend this line of research by analyzing bivariate almost stochastic dominance. Precisely, the stochastic dominance rule of Epstein and Tanny (1980) is weakened by considering the preferences shared by most (but not all) correlation averse decision makers with submodular utilities. Let us now provide some motivation for the almost version of bivariate stochastic dominance.1 In what follows, we concentrate on the dependence structure. Let us divide the real plane into four quadrants, where both outcomes are simultaneously large (upper right), one is large and the other small (upper left and lower right) and both are simultaneously small (lower left), respectively. The really adverse case is the last one (lower left). Now, consider two discrete (say) distributions with identical probabilities over the three quadrants but differing only on the last, lower left one, where one distribution expresses more positive dependence compared with the other (in the sense that its distribution function dominates the other one over this quadrant). As correlation averse decision makers prefer that the losses be as least correlated as possible, there is a clear 1 A detailed motivating example follows in Sect. 2.

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ranking of these two distributions, with a preference for the dominated distribution function expressing weaker positive dependence. Now, assume that we slightly modify the probability masses in the upper right quadrant, corresponding to the most favorable situation, making the two distributions not comparable anymore (in the sense that the distribution functions cross each other). This makes the whole distributions not comparable in stochastic dominance as all correlation averse decision makers do not agree anymore. However, provided the perturbation applied in the upper right quadrant is small enough, we may still prefer the one with the smaller correlation among simultaneous losses, i.e., in the lower left quadrant. This is the typical story behind the almost version of bivariate stochastic dominance. To reveal a preference for most decision makers, but not for all of them, we restrict the class of correlation averse (resp. loving) utility functions to a subset of it by confining correlation aversion (resp. loving) to some acceptable levels. We then study the new stochastic dominance rule corresponding to the common preferences of the decision makers exhibiting confined correlation aversion (resp. loving). We provide the integral condition for bivariate almost stochastic dominance which can be related to two-attribute utility functions excluding extreme preferences. The remainder of the paper is organized as follows. In Sect. 2, we study attitudes toward correlation and we introduce the concept of confined correlation aversion and correlation loving. Then, we examine the stochastic dominance rule expressing the common preferences of all the decision makers exhibiting confined correlation aversion or loving. Section 3 is devoted to some useful properties of confined correlation aversion or loving and the corresponding bivariate almost stochastic dominance rules. These results extend those obtained by Epstein and Tanny (1980) to the almost case. Section 4 discusses two applications illustrating the relevance of the approach developed in the present paper. The first one extends the study conducted in Denuit et al. (2011) and provides an application of our findings to the saving decision. The second one is devoted to the impact of envy and altruism on investment choices, considering the stochastic dominance framework developed by Kaplanski and Levy (2013), where the two attributes represent the decision maker’s wealth and the peer group’s wealth, respectively. Whereas the two-attribute utility function is still non-decreasing with respect to the investor’s wealth, it may increase (altruism) or decrease (envy) in the peer group’s wealth. Despite the apparent difference with our setting, we show that our results can easily be adapted to cover this case, too. Section 5 discusses related concepts. First, a comparison is performed with the tractable almost stochastic dominance approach proposed by Lizyayev and Ruszczynski (2012) and its possible extension to the bivariate case. Then, financial applications are discussed, and testing procedures are derived in the context of optimal portfolio selection. Section 6 briefly concludes the paper. The proofs of the main results are gathered in the appendix. 2 Confined correlation attitudes 2.1 Correlation aversion Consider pairs (X 1 , X 2 ) of commodities. Assume that the first attribute X 1 is valued in the interval [a1 , b1 ] and the second one X 2 is valued in the interval [a2 , b2 ]. For

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instance, X 1 and X 2 may represent consumption at two different times, the values of two assets at a future date or the wealth and health of an individual. In the expected utility setting, a decision maker with utility function u prefers (Y1 , Y2 ) over (X 1 , X 2 ) if E[u(X 1 , X 2 )] ≤ E[u(Y1 , Y2 )]. Given a utility function u of two attributes x1 and i+ j 1 ,x 2 ) x2 , we denote the cross derivatives as u (i, j) (x1 , x2 ) = ∂ u(x i j . We assume that (∂ x1 ) (∂ x2 )

the utility function u is non-decreasing, i.e., that u (1,0) ≥ 0 and u (0,1) ≥ 0, which simply means that both attributes are goods (except in Section 4.2 devoted to envy and altruism where we may have u (0,1) ≤ 0 expressing the fact that the utility of an envious person decreases with the peer group’s wealth). Correlation aversion as defined by Eeckhoudt et al. (2007) is related to the substitutability of goods first addressed by Richard (1975). More precisely, consider x1 ∈ [a1 , b1 ], x2 ∈ [a2 , b2 ], h 1 ∈ [0, b1 − x1 ] and h 2 ∈ [0, b2 − x2 ]. Define the lotteries L offering either (x1 , x2 ) or (x1 + h 1 , x2 + h 2 ) with equal probability 0.5 and M offering either (x1 , x2 + h 2 ) or (x1 + h 1 , x2 ) with equal probability 0.5. Eeckhoudt et al. (2007) require the preference of M over L whatever x1 , x2 , h 1 ≥ 0 and h 2 ≥ 0 for correlation aversion. In the expected utility setting, this means that the utility function has to fulfill u(x1 , x2 ) + u(x1 + h 1 , x2 + h 2 ) ≤ u(x1 , x2 + h 2 ) + u(x1 + h 1 , y2 )

(1)

for any x1 , x2 , h 1 ≥ 0 and h 2 ≥ 0. Condition (1) defines submodularity. If u is twice differentiable, then (1) is equivalent to u (1,1) ≤ 0. Henceforth, we denote as Uca the class of all the correlation averse utility functions, i.e., non-decreasing functions u satisfying (1). To be precise,

u ∈ Uca

⎧ (1,0) ≥0 ⎪ ⎨u (0,1) ⇔ u ≥0 ⎪ ⎩ (1,1) u ≤ 0.

While conditions u (1,0) ≥ 0 and u (0,1) ≥ 0 simply mean that both attributes are goods, condition u (1,1) ≤ 0 expresses correlation averse behavior. From Epstein and Tanny (1980), we know that E[u(X 1 , X 2 )] ≤ E[u(Y1 , Y2 )] for all u ∈ Uca if, and only if, the joint distribution functions of the random couples (X 1 , X 2 ) and (Y1 , Y2 ) satisfy Pr[X 1 ≤ t1 , X 2 ≤ t2 ] ≥ Pr[Y1 ≤ t1 , Y2 ≤ t2 ]

for all t1 and t2 .

(2)

We write (X 1 , X 2 ) ca (Y1 , Y2 ) if (2) holds. The bivariate stochastic dominance ca expresses the common preferences of all the correlation averse decision makers. Note that (2) ensures that Pr[X 1 ≤ t1 ] ≥ Pr[Y1 ≤ t1 ] for all t1 and Pr[X 2 ≤ t2 ] ≥ Pr[Y2 ≤ t2 ] for all t2 , that is, X i fsd Yi for i = 1, 2, where fsd stands for univariate first-order stochastic dominance. Let us provide the economic intuition behind the bivariate stochastic dominance rule ca generated by the common preferences of all correlation averse decision makers.

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Comparing the joint distribution functions means that the economic agent faced with pairs of attributes selects a target t1 for the first attribute and a target t2 for the second attribute and then ranks the pairs based on the joint probability that they do not reach these specified targets; see (2). The decision maker thus acts in order to minimize the probability of not reaching the target.

2.2 Confining correlation aversion: A simple motivating example Consider two risky assets with end-of-period values modeled by means of the independent random variables Y1 and Y2 assuming values -2, -1, 1 and 2 with equal probabilities 41 . Let us now modify this joint distribution by applying an elementary increasing correlation transformation in the lower left quadrant, producing positively correlated end-of-period values when losses affect both assets. Precisely, consider the random variables X 1 and X 2 distributed as Y1 and Y2 except when both components are negative, where Pr[X 1 = −2, X 2 = −2] = Pr[X 1 = −1, X 2 = −1] =

1 . 8

The probability masses placed at (−2, −1) and at (−1, −2) have thus been transferred to (−2, −2) and (−1, −1) so that when losses occur simultaneously, they are both “large” in the (−2, −2) case or both “small” in the (−1, −1) case. Given that losses occur for both assets, i.e., X 1 < 0 and X 2 < 0, asset values are now comonotonic, or perfectly positively dependent. As the joint distribution function for X 1 and X 2 dominates the one of Y1 and Y2 , any expected utility maximizer who is correlation averse in the sense of Epstein and Tanny (1980) and of Eeckhoudt et al. (2007) prefers the random couple (Y1 , Y2 ) over (X 1 , X 2 ), that is, (X 1 , X 2 ) ca (Y1 , Y2 ) holds true. Let us now consider another couple of random variables, Z 1 and Z 2 , say, distributed as Y1 and Y2 except in the upper right quadrant, where both assets generate gains. Precisely, we again apply a correlation increasing transformation so that 1 16 1 Pr[Z 1 = 1, Z 2 = 2] = Pr[Y1 = 1, Y2 = 2] − δ = 16 1 Pr[Z 1 = 2, Z 2 = 1] = Pr[Y1 = 2, Y2 = 1] − δ = 16 1 Pr[Z 1 = 2, Z 2 = 2] = Pr[Y1 = 2, Y2 = 2] + δ = 16 Pr[Z 1 = 1, Z 2 = 1] = Pr[Y1 = 1, Y2 = 1] + δ =

+δ −δ −δ +δ

1 for some positive δ less than 16 . As the respective joint distribution functions of (X 1 , X 2 ) and of (Z 1 , Z 2 ) cross, there is no unanimous preference for (Z 1 , Z 2 ) or for

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(X 1 , X 2 ) by all the correlation averse decision makers2 : some of them prefer (X 1 , X 2 ) over (Z 1 , Z 2 ) despite the maximal positive correlation between simultaneous losses with (X 1 , X 2 ). This is, for instance, the case for satisfiers selecting target values t1 and t2 for both assets, with t1 > −1 and t2 > −1, whose two-attribute utility function is of the form  1 if x1 > t1 and x2 > t2 (3) u(x1 , x2 ) = 0 otherwise. Those decision makers do not consider the simultaneous losses occurring in the lower left quadrant and are thus indifferent between (X 1 , X 2 ) and (Y1 , Y2 ), whereas they are hurt by the increase in correlation in the upper right quadrant so that they dislike the shift to (Z 1 , Z 2 ). As another example, we may consider a decision maker with kinked utility function u(x1 , x2 ) = (x1 + x2 )+ where (ξ )+ denotes the positive part of the real ξ , or more generally u(x1 , x2 ) = v((x1 + x2 )+ ) for some non-decreasing and concave function v. However, the authors, presumably as most decision makers, still prefer (Z 1 , Z 2 ) over (X 1 , X 2 ) at least for δ small enough, despite the deterioration of the joint distribution for simultaneous gains. Hence, moderate violations of correlation aversion may be allowed for real-world decision making. This is precisely the motivation for introducing an almost version of bivariate stochastic dominance, aiming to focus on the preferences of most decision makers by allowing for moderate violations of correlation aversion. Basically, if the joint distribution of the pair of assets is dominated by another one except over a small enough violation set and if the extent of the violation is limited then this pair of assets may still be preferred. To decide whether the violation is small enough to be neglected, we compare the “volume” comprised between the two joint distribution functions over the violation set to the total “volume.” Pre-   cisely, we compare the integral of  Pr[X 1 ≤ t1 , X 2 ≤ t2 ] − Pr[Z 1 ≤ t1 , Z 2 ≤ t2 ] over the whole domain [−2, 2] × [−2, 2] to the corresponding integral over the violation set and we require that the former does not exceed a fixed percentage of the latter. In our example, we see that the joint distribution function of the pair (Z 1 , Z 2 ) dominates the one of (X 1 , X 2 ) over the violation set [1, 2] × [1, 2] so that the volume comprised between the two surfaces over this square is δ 2 . If δ is small enough, then the extent to which correlation aversion is violated may be considered as small enough and the decision maker may still prefer (Z 1 , Z 2 ). To appreciate whether the extent to which the dominance relation is violated can be considered as being small enough, we compare it to the total volume comprised between the two surfaces, which is equal to 1 16δ 2 2 16 + δ . As long as the decision maker is ready to accept violations less than 1+16δ 2 of the total volume, the pair (Z 1 , Z 2 ) is still preferred. For instance, if the tolerance is up to 5 %, then (Z 1 , Z 2 ) is preferred over (X 1 , X 2 ) as long as δ ≤ 5.74 %. 2 In other words, (X , X ) and (Z , Z ) cannot be ordered with respect to  . In such a case, increasing ca 1 2 1 2

the order of the bivariate stochastic dominance rule may be an option, as discussed in Denuit and Mesfioui (2012).

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2.3 Confined correlation aversion The class Uca of correlation averse utilities includes some extreme forms of preferences, i.e., ca cannot rank some lotteries that most of the correlation averse individuals can easily choose. To overcome this drawback, we can appropriately constrain the partial derivative u (1,1) , in the spirit of the univariate almost stochastic dominance introduced by Leshno and Levy (2002). This leads to the concept of confined correlation aversion, which is precisely defined as follows. ε contains all Definition 1 Let ε be a real constant such that 0 < ε < 21 . The class Uca the elements of Uca satisfying

0 ≤ −u (1,1) (x1 , x2 ) ≤ inf{−u (1,1) }



1 −1 ε

for all x1 and x2 .

(4)

Such a utility function is said to express ε-confined correlation aversion. By controlling ε, Eq. (4) determines how to exclude some extreme preferences. For example, the utility function (3) can be excluded by condition (4) as ε > 0. Let us now consider the common preferences of all the ε-confined correlation averse decision makers. To this end, let us define the following new bivariate stochastic dominance rule, which allows for some moderate violations of condition (2) corresponding to ca , and is referred to as “almost correlation averse stochastic dominance”. Recall that first-degree stochastic dominance rule is denoted as fsd . Definition 2 (Almost correlation averse stochastic dominance) Assume that (X 1 , X 2 ) ca (Y1 , Y2 ) does not hold and define the violation set S of condition (2) as the set of all (x1 , x2 ) such that Pr[X 1 ≤ x1 , X 2 ≤ x2 ] < Pr[Y1 ≤ x1 , Y2 ≤ x2 ]. Then, (X 1 , X 2 ) εca (Y1 , Y2 ) holds if X 1 fsd Y1 , X 2 fsd Y2 and



Pr[Y1 ≤ x1 , Y2 ≤ x2 ] − Pr[X 1 ≤ x1 , X 2 ≤ x2 ] dx1 dx2

S

b1 b2     ≤ε  Pr[X 1 ≤ x1 , X 2 ≤ x2 ] − Pr[Y1 ≤ x1 , Y2 ≤ x2 ]dx1 dx2 .

(5)

a1 a2

In words, condition (5) means that the violation set S and the extent to which (2) is violated on S must be moderate enough to ensure that the integral over S of the difference in the respective joint distribution functions is smaller than ε times the total volume confined between these distribution functions. ε . It states that The following result connects εca to the utility functions in Uca ε ca expresses the common preferences of all the decision makers with an ε-confined correlation averse utility function. ε . Theorem 1 (X 1 , X 2 ) εca (Y1 , Y2 ) ⇔ E[u(X 1 , X 2 )] ≤ E[u(Y1 , Y2 )] for all u in Uca

The proof of Theorem 1 is given in Appendix A.

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2.4 Confined correlation loving Decision makers may not be correlation averse. For example, an individual might think that money is less valuable when leisure time is limited. These types of individuals are called correlation loving. According to Eeckhoudt et al. (2007), a correlation lover would prefer lottery L to M which are mentioned in the previous section, i.e., u(x1 , x2 ) + u(x1 + h 1 , x2 + h 2 ) ≥ u(x1 , x2 + h 2 ) + u(x1 + h 1 , y2 )

(6)

for any x1 , x2 , h 1 ≥ 0 and h 2 ≥ 0. Condition (6) defines supermodularity. If u is twice differentiable, then (6) is equivalent to u (1,1) ≥ 0. Let us denote Ucl as the class of all the correlation loving utility functions, i.e., non-decreasing functions u satisfying (6). Precisely, ⎧ (1,0) ≥0 ⎪ ⎨u (0,1) u ∈ Ucl ⇔ u ≥0 ⎪ ⎩ (1,1) u ≥ 0. Obviously, as we demonstrated in the case of Uca , some preferences in Ucl could be too extreme and should be excluded. Let us then define the concept of confined correlation loving as follows. φ

Definition 3 Let φ be a real constant such that 0 < φ < 21 . The class Ucl contains all the elements of Ucl satisfying 0 ≤ u (1,1) (x1 , x2 ) ≤ inf{u (1,1) }



1 −1 φ

for all x1 and x2 .

(7)

Such a utility function is said to express φ-confined correlation loving. Condition (7) requires that the ratio of the maximum value to the minimum value of u (1,1) be bounded. Thus, it can exclude some extreme preferences. Let cl express the preferences of all the correlation loving decision makers. Denuit et al. (1999) indicate that (Y1 , Y2 ) dominates (X 1 , X 2 ) in the sense of bivariate (1,1)convex stochastic dominance, which is here denoted as (X 1 , X 2 ) cl (Y1 , Y2 ), if E[u(X 1 , X 2 )] ≤ E[u(Y1 , Y2 )] for all u ∈ Ucl . It is easy to find that the necessary and sufficient condition for (X 1 , X 2 ) cl (Y1 , Y2 ) to hold is Pr[X 1 > t1 , X 2 > t2 ] ≤ Pr[Y1 > t1 , Y2 > t2 ]

for all t1 and t2 .

(8)

Condition (8) requires that the joint survival, or excess function of (Y1 , Y2 ) is always larger than the corresponding function of (X 1 , X 2 ), indicating that it is more likely that Y1 and Y2 simultaneously assume large values compared to X 1 and X 2 .3 The 3 The terminology survival function refers to survival analysis in biostatistics. Here, we prefer the name

excess function since the probabilities of exceeding the levels t1 and t2 are involved.

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following new stochastic dominance rule, which is referred to as “almost correlation loving stochastic dominance,” allows for some moderate violations of the condition (8) corresponding to cl . Definition 4 (Almost correlation loving stochastic dominance) Assume that (X 1 , X 2 ) cl (Y1 , Y2 ) does not hold and define the violation set Sˆ of condition (8) as the set of all (x1 , x2 ) such that Pr[X 1 > x1 , X 2 > x2 ] > Pr[Y1 > x1 , Y2 > x2 ]. Then, φ (X 1 , X 2 ) cl (Y1 , Y2 ) holds if X 1 fsd Y1 , X 2 fsd Y2 and



Pr[X 1 > x1 , X 2 > x2 ] − Pr[Y1 > x1 , Y2 > x2 ] dx1 dx2

Sˆ

b1 b2     ≤φ  Pr[X 1 > x1 , X 2 > x2 ] − Pr[Y1 > x1 , Y2 > x2 ]dx1 dx2 .

(9)

a1 a2

In words, condition (9) still means that the violation set Sˆ must be moderate enough to ensure that the integral over Sˆ of the difference in the respective joint excess functions is smaller than φ times the total volume confined between these excess functions. φ φ The following result further connects cl to the utility functions in Ucl . φ

φ

Theorem 2 (X 1 , X 2 ) cl (Y1 , Y2 ) ⇔ E[u(X 1 , X 2 )] ≤ E[u(Y1 , Y2 )] for all u in Ucl . φ

The proof of Theorem 2 is provided in Appendix B. Theorem 2 states that cl expresses the preferences of all the decision makers with an φ-confined correlation loving utility function. 3 Some properties 3.1 Identical marginals Note that correlation aversion has been defined by comparing two lotteries that are marginally equivalent, in the sense that under both L and M, the individual obtains x1 or x1 + h 1 with the same probability and x2 or x2 + h 2 with the same probability. The difference between L and M lies in the combination of the two goods: the correlation averse decision maker prefers a mix of a favorable together with an unfavorable case over two favorable or unfavorable outcomes arising simultaneously. Assuming that the pairs of attributes have identical univariate marginal distributions means that they behave identically in isolation but differ in the way they interact, i.e., in their respective dependence structures. This assumption allows us to concentrate on the correlation structure only, without marginal effects. Additional results can be obtained in this more restrictive setting. The next result shows that εca still induces a preference for low correlation as (X 1 , X 2 ) εca (Y1 , Y2 ) implies that the attributes (X 1 , X 2 ) are more correlated than the

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attributes (Y1 , Y2 ) when the univariate marginals are identical, i.e., when the identities Pr[X 1 ≤ t] = Pr [Y1 ≤ t] and Pr[X 2 ≤ t] = Pr [Y2 ≤ t] hold for all t.

(10)

Property 1 If (X 1 , X 2 ) and (Y1 , Y2 ) have the same univariate marginals, that is, condition (10) holds true, then (X 1 , X 2 ) εca (Y1 , Y2 ) ⇒ Cov[X 1 , X 2 ] ≥ Cov[Y1 , Y2 ] and φ

(X 1 , X 2 ) cl (Y1 , Y2 ) ⇒ Cov[X 1 , X 2 ] ≤ Cov[Y1 , Y2 ]. The proof of Property 1 is given in Appendix C. Property 1 shows that any εca ranking implies a preference for less correlated attributes when the marginals are identical. The restriction to identical marginals may seem artificial but appears to be useful to isolate the effect of the dependence structure. The comparison to the independent case is particularly instructive in many cases. Assume for instance that the decision maker is offered two alternate jobs located in different cities and that income and climate are the two attributes. If the climate and the income are similar (so that (10) holds true, for instance, because of the kind of position offered and the geographic location of the cities), the difference may be in the pairing of these two variables. For instance, the salary may be increased in case of a very cold winter or the salary may be lost in case climatic events make transportation to the office impossible. 3.2 Aggregation to a single position When the attributes are of a financial nature, for example, and are aggregated in a single position, we denote as v a utility function of the single resulting attribute. Henceforth, we denote as ssd the univariate second-order stochastic dominance. Furthermore, denoting as v (k) the kth derivative of the single-attribute utility function v, denote

 Ufsd = single-attribute utility functions v|v (1) ≥ 0

 Ussd = single-attribute utility functions v|v (1) ≥ 0 and v (2) ≤ 0 .

(11) (12)

Clearly, X fsd Y (resp. X ssd Y ) if, and only if, E[v(X )] ≤ E[v(Y )] for all u ∈ Ufsd (resp. u ∈ Ussd ). We know from Epstein and Tanny (1980) that the ranking (X 1 , X 2 ) ca (Y1 , Y2 ) allows us to compare linear combinations w1 X 1 + w2 X 2 and w1 Y1 + w2 Y2 made of (X 1 , X 2 ) and (Y1 , Y2 ) with nonnegative coefficients w1 and w2 in second-order stochastic dominance. To be precise, we have (X 1 , X 2 ) ca (Y1 , Y2 ) ⇒ w1 X 1 +w2 X 2 ssd w1 Y1 +w2 Y2 ,

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This result shows that when all the correlation averse decision makers prefer (Y1 , Y2 ) over (X 1 , X 2 ), any linear combination of Y1 and Y2 dominates in second-order stochastic dominance the corresponding linear combination of X 1 and X 2 . Let us now establish that a similar result holds for εca and ε-almost second-order stochastic dominance defined by Tzeng et al. (2013). As explained in Leshno and Levy (2002), Ussd contains some extreme utility functions that presumably rarely represent real-world investors’ preferences. The prototype is v(x) = min{x, r } for some constant r . To reveal a preference for most investors, but not for all of them, Leshno and Levy (2002) further impose restrictions on the utility function and define     1

 ε − 1 for all x , Ussd = v ∈ Ussd  − v (2) (x) ≤ inf −v (2) ε

(13)

  where ε ∈ 0, 21 . Now, given two random variables X and Y, X εssd Y ⇔ ε . The next E[v(X )] ≤ E[v(Y )] for every single-attribute utility function v in Ussd ε ε result relates ca to ssd as ca is known to be related to ssd from Epstein and Tanny (1980). Property 2 (X 1 , X 2 ) εca (Y1 , Y2 ) ⇒ w1 X 1 + w2 X 2 εssd w1 Y1 + w2 Y2 for all w1 and w2 ≥ 0. The proof of Property 2 is given in Appendix D. φ Now, let us further examine whether the ranking (X 1 , X 2 ) cl (Y1 , Y2 ) can help us to compare linear combinations w1 X 1 + w2 X 2 and w1 Y1 + w2 Y2 . Following Tzeng et al. (2013), we first derive the condition to rank distributions for most risk lovers, which is defined as follows    1

 φ − 1 for all x , Url = v ∈ Ufsd 0 ≤ v (2) (x) ≤ inf v (2) φ   φ where φ ∈ 0, 21 . Furthermore, define X rl Y as E[X ] ≤ E[Y ], and

b 

b  b     Pr[X > t] − Pr[Y > t] dtdx ≤ φ  Pr[X > t] − Pr[Y > t] dt dx,

 x

a

x

(14) b b where  denotes the set of x such that x Pr[X > t]dt > x Pr[Y > t]dt. The φ φ following result connects rl with v in Url . φ

φ

Theorem 3 X rl Y ⇔ E[v(X )] ≤ E[v(Y )] for all v in Url . The proof of Theorem 3 is given in Appendix E. Note that when φ approaches zero, Eq. (14) holds if, and only if,  is an empty set, b b i.e., x Pr[X > t]dt ≤ x Pr[Y > t]dt for all x. This inequality defines the increasing convex order (see Shaked and Shanthikumar 2007), also called the stop-loss order in

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actuarial sciences (see Denuit et al. 2005). Furthermore, the set of Url is equivalent to Url when φ approaches zero. Therefore, Theorem 3 predicts that E[v(X )] ≤ E[v(Y )] for all v in Url if and only if

b

b Pr[X > t]dt ≤

x

Pr[Y > t]dt

for all x,

Pr[Y ≤ t]dt

for all x.

x

or, equivalently, if and only if

b

b Pr[X ≤ t]dt ≥

x

x

This is consistent with the second-degree stochastic dominance rule for risk lovers in Levy and Wiener (1998). The following property relates the univariate and bivariate almost stochastic dominance rules for correlation and risk lovers. φ

φ

Property 3 (X 1 , X 2 ) cl (Y1 , Y2 ) ⇒ w1 X 1 + w2 X 2 rl w1 Y1 + w2 Y2 for all w1 and w2 ≥ 0. The proof of Property 3 is given in Appendix F. 4 Illustrations 4.1 An application to the saving problem In this section, we illustrate the results derived in the preceding sections by analyzing the saving problem. The effect of risk on the saving decision is essential to understanding the intertemporal behavior of consumption. This topic has received much attention in the literature, see, e.g., Kimball (1990), Courbage and Rey (2007), and Menegatti (2009). Eeckhoudt and Schlesinger (2008) have adopted a univariate model and have examined how a deterioration in the sense of nth-degree stochastic dominance in the background risk changes the optimal saving. Denuit et al. (2011) use a bivariate model to examine the effect of an increase in the joint distribution of the background financial and non-financial risks in the sense of bivariate higher-order increasing concave stochastic dominance on the optimal saving. Our findings can extend the literature to understand the relationship between a change in risk in terms of almost bivariate stochastic dominance and the optimal saving. In a two-period model, assume that the decision maker with initial wealth w0 and health condition h 0 can decide to save α in a risk-free asset, which will generate a positive return r in the second period. In the second period, the decision maker faces a financial background risk Y1 and a health background risk Y2 . Let u 0 and u 1 denote, (1,0) respectively the utility function in the first and second periods with u i ≥ 0, i = 0, 1. Thus, the objective function is

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389

  max u 0 (w0 − α, h 0 ) + E u 1 (Y1 + α (1 + r ) , Y2 ) . α

The first-order condition of the problem is (1,0)

− u0

  (1,0) (w0 − α, h 0 ) + (1 + r ) E u 1 (Y1 + α (1 + r ) , Y2 ) = 0.

(15)

(2,0)

Assume that u i ≤ 0, i = 0, 1, so that the second-order condition holds. Let αY denote the optimal saving. Assume that the pair of background risks (Y1 , Y2 ) experience a deterioration in terms of almost correlation averse stochastic dominance and become (X 1 , X 2 ). Let α X denote the optimal saving under (X 1 , X 2 ). Since the second-order condition holds, we have α X ≥ αY if, and only if, (1,0)

−u 0

  (1,0) (w0 − αY , h 0 ) + (1 + r ) E u 1 (X 1 + αY (1 + r ) , X 2 ) ≥ 0.

From Eq. (15), the above equation can be written as     (1,0) (1,0) E −u 1 (X 1 + αY (1 + r ) , X 2 ) ≤ E −u 1 (Y1 + αY (1 + r ) , Y2 ) . (16) According to Theorem 1, we know that (X 1 , X 2 ) εca (Y1 , Y2 ) ensures that inequalε . Note that −u (1,0) ∈ U ε ity (16) is valid for all utilities u 1 such that −u (1,0) ∈ Uca ca 1 1 means that (2,0)

(1,1)

≤ 0, u 1

(2,1)

≤ 0, u 1 ≥ 0 and  1 (2,1) (2,1) −1 for all x1 and x2 . u 1 (x1 , x2 ) ≤ inf{u 1 } ε u1

(17)

Thus, Theorem 1 helps us to conclude that α X ≥ αY for all u 1 satisfying conditions in (17) if, and only if, (X 1 , X 2 ) εca (Y1 , Y2 ). By the same token, Theorem 2 indicates that α X ≥ αY for all u 1 satisfying the following conditions u (2,0) ≤ 0, u 1(1,1) ≤ 0, u (2,1) ≤ 0 and 1 1 1 (2,1) (2,1) −u 1 (x1 , x2 ) ≤ inf{−u 1 } −1 for all x1 and x2 , φ

(18)

φ

if, and only if, (X 1 , X 2 ) cl (Y1 , Y2 ). 4.2 Income distributions with envy and altruism Since Atkinson (1970), the literature has provided many insightful findings in ranking income distributions. However, most contributions share a common assumption that

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the welfare function only depends on individuals’ income but not on that of the peer group. Relaxing this assumption, the preference of keeping up with the Joneses has been well analyzed in the recent literature, see, e.g., Abel (1990), Dupor and Liu (2003), Garcia-Penalosa and Turnovsky (2006; 2008), and Barnett et al. (2010). In this section, we analyze the effect of envy and altruism on income with the help of bivariate stochastic dominance rules, after Kaplanski and Levy (2013). Assume that the first attribute represents the investor’s wealth, whereas the second attribute corresponds to the peer group’s wealth. Clearly, the individual always prefers to have more rather than less wealth so that the two-attribute utility function u representing his preferences fulfills u (1,0) ≥ 0. However, the individual may be an envious person who is happy if the peer group has less wealth so that u (0,1) ≤ 0. Alternatively, he may be altruistic with u (0,1) ≥ 0. Thus, the first derivative with respect to the second attribute is not necessarily positive, contrarily to the assumptions retained in the preceding sections. Considering the interactions between the individual’s wealth and the peer group’s wealth, some individuals may prefer relatively large incomes when the peer group also has a high income, which corresponds to correlation loving and manifests in u (1,1) ≥ 0. Correlation aversion u (1,1) ≤ 0 then corresponds to the preference for relatively large individual incomes when the peer group has a small income. To sum up, envy (u (0,1) ≤ 0) or altruism (u (0,1) ≥ 0) combined with correlation loving (u (1,1) ≥ 0) or correlation aversion (u (1,1) ≤ 0) may guide the choices made by an individual. Let us now explain how to adapt the results derived earlier in this paper to cover the envy–altruism case examined by Kaplanski and Levy (2013). To fix the ideas, consider a two-attribute utility function u expressing envy and correlation loving, i.e., such that u (1,0) ≥ 0, u (0,1) ≤ 0 and u (1,1) ≥ 0.

(19)

Then, E[u(X 1 , X 2 )] ≤ E[u(Y1 , Y2 )] for all utilities fulfilling (19) ⇔ E[u(X 1 , −(−X 2 ))] ≤ E[u(Y1 , −(−Y2 ))] for all utilities fulfilling (19) ⇔ E[v(X 1 , −X 2 )] ≤ E[v(Y1 , −Y2 )] for all utilities v such that v (1,0) ≥ 0, v (0,1) ≥ 0 and v (1,1) ≤ 0 as the function v defined from u by v(x1 , x2 ) = u(x1 , −x2 ) has derivatives v (1,0) = u (1,0) ≥ 0, v (0,1) = −u (0,1) ≥ 0 and v (1,1) = −u (1,1) ≤ 0. So, we are back to the kind of dominance relation considered in the preceding sections but for (X 1 , −X 2 ) and (Y1 , −Y2 ), i.e., replacing gains with losses for the peer group. As a particular example of special interest, consider the joint probability distributions described in Tables 1, 2 and 3 for some income distribution together with the reference income distribution. Table 1 is the case where the income assumes the values 10, 20 and 30 with the respective probabilities 0.3, 0.4 and 0.3. The reference income assumes values −30, −20 and −10 with respective probabilities 0.3, 0.4 and 0.3. In Table 1, individuals only envy their own groups, respectively. In Table 2, the income distribution remains the same as in Table 1, but individuals change the way they envy others. Rich people envy the poor ones while the poor ones envy the rich ones, but the

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Bivariate almost stochastic dominance Table 1 Joint probability distribution of individual and reference incomes. Case 1

Table 2 Joint probability distribution of individual and reference incomes. Case 2

Table 3 Joint probability distribution of individual and reference incomes. Case 3

Joint probabilities

391 Income = 10 Income = 20 Income = 30

Reference income = −30 0.0

0.0

0.3

Reference income = −20 0.0

0.4

0.0

Reference income = −10 0.3

0.0

0.0

Joint probabilities

Income = 10 Income = 20 Income = 30

Reference income = −30 0.3

0.0

0.0

Reference income = −20 0.0

0.4

0.0

Reference income = −10 0.0

0.0

0.3

Joint probabilities

Income = 10 Income = 20 Income = 30

Reference income = −30 0.1

0.0

0.0

Reference income = −20 0.0

0.4

0.0

Reference income = −10 0.0

0.0

0.5

middle class still envy their own groups. In this case, although the reference income distribution does not change, the society is obviously worse off with respect to bivariate stochastic dominance. This case demonstrates that simply a change in how people envy others could reduce social welfare. Table 3 is the case where the income distribution is improved while individuals change the way they envy others. The income now assumes values 10, 20 and 30 with respective probabilities 0.1, 0.4 and 0.5. In this case, although the income distribution is improved with respect to univariate first-degree stochastic dominance, we cannot conclude that the social welfare is improved with respect to bivariate stochastic dominance. But, employing the concept of almost bivariate stochastic dominance, we can show that the society becomes better off if ε ≥ 1/3. This case indicates that if the government can do something to improve the income distribution while people change their reference groups, almost bivariate stochastic dominance can help the social planner to recognize cases where the social welfare is greater. 5 Discussion and conclusion This section discusses alternative definitions of bivariate almost stochastic dominance and explains how to test for such dominance rules in financial problems. 5.1 Tractable bivariate almost stochastic dominance Another version of almost stochastic dominance has been proposed by Lizyayev and Ruszczynski (2012). Considering the first-order case, these authors suggest in their Definition 3 that a random variable Y ε -dominates a random variable X if there exists a nonnegative random variable Z such that X fsd Y + Z holds and Z is small enough, in the sense that E[Z ] ≤ ε. Lizyayev and Ruszczynski (2012) further provide

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algorithms to implement their newly defined almost stochastic dominance concept and demonstrate that their definition is computationally tractable. Let us now turn to the bivariate case. It is tempting to define the ε-almost stochastic dominance of (Y1 , Y2 ) over (X 1 , X 2 ) by the existence of a nonnegative, small enough random variable Z such that (X 1 , X 2 ) ca (Y1 + Z , Y2 + Z ) holds. Assume that the restriction placed on Z is that its distribution puts a high enough probability mass on 0, i.e., that Pr[Z > 0] ≤ ε holds. Let I [A] be the indicator for event A, i.e., I [A] = 1 if A is realized and I [A] = 0 otherwise. We then have I [Y1 + Z ≤ t1 , Y2 + Z ≤ t2 ] ≥ I [Y1 ≤ t1 , Y2 ≤ t2 , Z = 0] = I [Y1 ≤ t1 , Y2 ≤ t2 ](1 − I [Z > 0]) = I [Y1 ≤ t1 , Y2 ≤ t2 ] − I [Y1 ≤ t1 , Y2 ≤ t2 ]I [Z > 0] ≥ I [Y1 ≤ t1 , Y2 ≤ t2 ] − I [Z > 0] so that Pr[Y1 + Z ≤ t1 , Y2 + Z ≤ t2 ] ≥ Pr[Y1 ≤ t1 , Y2 ≤ t2 ] − ε. As Z is assumed to be such that (X 1 , X 2 ) ca (Y1 + Z , Y2 + Z ) holds true, we also have Pr[X 1 ≤ t1 , X 2 ≤ t2 ] ≥ Pr[Y1 + Z ≤ t1 , Y2 + Z ≤ t2 ] so that Pr[X 1 ≤ t1 , X 2 ≤ t2 ] ≥ Pr[Y1 ≤ t1 , Y2 ≤ t2 ] − ε holds. However, adding the comonotonic pair (Z , Z ) to (Y1 , Y2 ) simultaneously increases the size of each component, which is beneficial when both attributes are goods, but may also reinforce their dependence as long as Z is positively correlated with Y1 and Y2 , which tends to deteriorate the situation for correlation averse decision makers. This is easily seen from Cov[Y1 + Z , Y2 + Z ] = Cov[Y1 , Y2 ] + Cov[Y1 , Z ] + Cov[Y2 , Z ] + V ar [Z ] which exceeds Cov[Y1 , Y2 ] when Y1 and Y2 are positively related to, or independent of, Z . In the univariate case, the joint distribution of Y and Z does not matter as fsd only considers the respective sizes of the random variables to be compared and Y ≤ Y + Z holds with probability 1: Adding the nonnegative Z to Y has thus a clear effect in terms of fsd . This is no more true in the bivariate case where the addition of Z to both components of the random couple may have an ambiguous effect with respect to ca : making both components larger, which is beneficial, but also possibly more correlated, which is detrimental for correlation averse decision makers. Therefore, some additional constraints have to be imposed on the joint distribution of Y1 , Y2 and Z in the bivariate case, which makes the construction less appealing. The same comments apply to the case where the nonnegative pair (Z 1 , Z 2 ) would be added to (Y1 , Y2 ). We could nevertheless mimic Definition 2 in Lizyayev and Ruszczynski (2012) and define the “tractable ε-almost stochastic dominance” of (Y1 , Y2 ) over (X 1 , X 2 ) by the condition Pr[Y1 ≤ t1 , Y2 ≤ t2 ] − Pr[X 1 ≤ t1 , X 2 ≤ t2 ] ≤ ε

for all t1 , t2

(20)

with the additional condition that (Y1 , Y2 ) ca (X 1 , X 2 ) does not hold, i.e., there exist t1 and t2 such that Pr[Y1 ≤ t1 , Y2 ≤ t2 ] < Pr[X 1 ≤ t1 , X 2 ≤ t2 ]. The second condition prevents situations where

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0 ≤ Pr[Y1 ≤ t1 , Y2 ≤ t2 ] − Pr[X 1 ≤ t1 , X 2 ≤ t2 ] ≤ ε

for all t1 , t2 ,

so that (Y1 , Y2 ) ca (X 1 , X 2 ) but (Y1 , Y2 ) simultaneously dominates (X 1 , X 2 ).4 Let us now discuss the link between this definition and our Definition 3 for almost correlation averse stochastic dominance εca . On the violation set S, we have 0 ≤ Pr[Y1 ≤ t1 , Y2 ≤ t2 ] − Pr[X 1 ≤ t1 , X 2 ≤ t2 ] ≤ ε so that



  Pr[Y1 ≤ t1 , Y2 ≤ t2 ] − Pr[X 1 ≤ t1 , X 2 ≤ t2 ] dt1 dt2 ≤ ε|S| (21) S

where |S| is the area of S. Hence, (X 1 , X 2 ) εca (Y1 , Y2 ) holds provided

b1 b2 |S| ≤

   Pr[X 1 ≤ t1 , X 2 ≤ t2 ] − Pr[Y1 ≤ t1 , Y2 ≤ t2 ]dt1 dt2 .

(22)

a1 a2

5.2 Testing financial decisions for bivariate almost stochastic dominance One important application of stochastic dominance in finance is to compute the efficient allocation in the portfolio selection problem.5 This section provides algorithms for the computation under different definitions of bivariate almost stochastic dominance. Suppose that in a portfolio problem, the investors face not only an investment risk but also a background risk. For simplicity, let us assume that the background risk is of financial nature. Assume that there exist N risky assets. We use here the empirical return distribution corresponding to the last T observation periods, assuming that each of the T historical returns has an equal probability T1 to occur. Thus, the return for asset i can be represented by the vector  Ri = ri1 , ri2 , . . . , riT where rit denotes the observation of return for asset i at time t. Furthermore, let B denote the vector representing the background risk where B = (b1 , b2 , . . . , b T ) and bt is the observation of the background risk in period t.  Assume that the investor  N can choose the investment weights τ = (τ1 , τ2 , . . . , τ N ), where τi ∈ [0, 1] and i=1 τi = 1. Therefore, the portfolio return of the investor with the background risk can be expressed as pt =

N 

rit τi + bt .

i=1 4 In the univariate case, it is also necessary to check whether X dominates Y in terms of stochastic dominance before examining whether Y dominates X in terms of the almost stochastic dominance concept proposed by Lizyayev and Ruszczynski (2012). 5 For example, please see Post (2003), Kuosmanen (2004) and Lizyayev (2012a).

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 1 2 T Sort p t and rename it as z t where z 1 < z 2 < · · · < z T . Denote  1 Z2 = (z ,Tz , . . . , z ).    In a similar way, Ri and B can be rearranged into X i = xi , xi , . . . , xi and Y = (y 1 , y 2 , . . . , y T ), respectively, where xit and y t are the return and background risk corresponding to z t . For the following tests, we examine whether the portfolio τ 0 = (τ01 , τ02 , . . . , τ0N ) with payoff Z0 = (z 01 , z 02 , . . . , z 0T ) is efficient. In other words, we check whether there exist other payoff vectors dominating Z0 .

5.2.1 Testing for bivariate almost stochastic dominance Since the background risk is financial, the two attributes of the utility function can be aggregated in a single position. Property 2 shows that we then have to test for εssd . Post (2003) used the following piecewise linear functions v(z t ) = α t + β t z t to represent the utility functions in the set of Ussd as defined in (12), where α t−1 + β t−1 z t−1 = α t + β t z t−1

for all t

(23)

and β 1 ≥ · · · ≥ β T ≥ 0. Note that restriction (23) ensures that the utility function v is continuous. Also note that β 1 ≥ · · · ≥ β T ≥ 0 is due to v (1) ≥ 0 and v (2) ≤ 0. Following Post (2003), we now consider the set of decision makers with utility function ε as defined in (13). Thus, additional conditions are imposed on β t to switch in Ussd ε . from Ussd to Ussd On basis of Eq. (11) in Post (2003), the following program provides a test for portfolio efficiency with respect to ε-almost second-order stochastic dominance εssd : minβ t θ under the constraints T  1  t t β z 0 − xit + θ ≥ 0 T

for all i

t=1

β1 ≥ · · · ≥ βT ≥ 0  β t − β t−1 1 − 1 δ≤ t ≤ δ ε z 0 − z 0t−1

for all t.

The last constraint that condition −v (2) (z) ≤  1  imposed in this program ensures ε (2) inf −v ε − 1 is satisfied for all z, as required in Ussd . This formulation is inspired from Eq. (19) in Post and Kopa (2013). Now, the portfolio τ 0 is εssd -efficient if, and only if, θ = 0, which is similar to Theorem 2 in Post (2003). 5.2.2 Testing for tractable bivariate almost stochastic dominance In addition to the testing procedure described above, let us now propose a test for the alternative almost stochastic dominance defined in (20) in this particular case. In a first stage, we need to check whether the evaluated portfolio τ 0 is efficient in terms of

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second-order stochastic dominance ssd . As proposed by Post (2003), this test can be performed as follows: minβ t θ under the constraints T  1  t t β z 0 − xit + θ ≥ 0 T

for all i

t=1

β 1 ≥ · · · ≥ β T ≥ 0. Second, for any ssd -efficient portfolio that passes the above test, we further check whether it is efficient in terms of the alternative almost stochastic dominance. Adopting the test proposed by Lizyayev and Ruszczynski (2012) in their Eqs. (10a)–(10e) gives the following optimization program:  N  T 1   t max xi wi + y t wi T t=1

i=1

under the constraints N 

xit wi + y t + q t,s ≥ z 0s

for s, t = 1, . . . , T

i=1 T 1  t,s (2)   q ≤ FZ z 0s + ε T

for s = 1 . . . , T

t=1

q t,s ≥ 0 N 

for s, t = 1, . . . , T

wi = 1

i=1

1 ≥ wi ≥ 0

for i = 1, . . . , N ,

where ε is the parameter of the tractable ε-almost second-order stochastic dominance z (2) rule proposed by Lizyayev and Ruszczynski (2012) and FZ (z) = a FZ (x)dx. If the maximum value of the above linear programming is larger than the mean of the evaluated portfolio τ 0 , then the evaluated portfolio is not efficient in terms of the alternative definition (20) of almost stochastic dominance. 5.2.3 Stocks and bonds in the long run Bali et al. (2009), Levy (2009) and Lizyayev and Ruszczynski (2012) have adopted the univariate almost stochastic dominance rules to evaluate the performance of stocks and bonds in the long run. The two algorithms described in Sects. 5.2.1 and 5.2.2 can help to identify the preference between stocks and bonds when there exists a financial background risk. To do so, the first step is to collect the empirical distributions of stocks, bonds and the background risk at various investment time horizons. The evaluated portfolio τ 0 could consist in investing 100 % in stocks, 100 % in bonds or a mix of stocks and

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bonds. The next step is to apply the above algorithms to examine whether the evaluated portfolio is efficient under different thresholds ε. If the background risk is financial, then the problem reduces to univariate almost stochastic dominance covered by Lizyayev and Ruszczynski (2012). Thus, adding a financial background risk in the examples treated by these authors could also show when almost correlation averse stochastic dominance and tractable bivarate almost stochastic dominance lead to similar or to different conclusions. 6 Conclusion In this paper, the concepts of correlation aversion and correlation loving have been extended to ε-confined correlation aversion and φ-confined correlation loving, respectively. We have defined bivariate almost stochastic dominance to allow for small violations of the condition defining the original concepts for confined correlation averse and confined correlation loving preferences. The impact of a change in risk in terms of bivariate almost stochastic dominance on optimal saving has been analyzed as an application of our findings. The extension to confine higher-order cross derivatives and to define the corresponding bivariate almost higher-degree stochastic dominance rules may be fruitful for future studies. Lizyayev (2012b) provides a thorough summary for the categories of secondorder stochastic dominance programs: majorization, distribution-based and revealed preference-type. In Sect. 5.2.1, we adopt a revealed preference-type program. On the other hand, Sect. 5.2.2 employs a distribution-based program. Lizyayev and Ruszczynski (2012) provide programs for both majorization and distribution-based to test for the almost stochastic dominance rule they proposed in the univariate case. Alternative programs to test for bivariate almost stochastic dominance rules certainly deserve interest in future studies. Also, empirical studies examining the efficient allocation under different versions of bivariate almost stochastic dominance in the presence of non-financial background risk are a promising research topic. 7 Appendix: Proofs of the main results 7.1 A Proof of Theorem 1 Let us start with the “⇒” part. Integration by parts shows that

b1 E[u(X 1 , X 2 )] = u(b1 , b2 ) −

u (1,0) (x1 , b2 ) Pr[X 1 ≤ x1 ]dx1

a1

b2 −

u (0,1) (b1 , x2 ) Pr[X 2 ≤ x2 ]dx2

a2

b1 b2 + a1 a2

123

u (1,1) (x1 , x2 ) Pr[X 1 ≤ x1 , X 2 ≤ x2 ]dx1 dx2 .

(24)

Bivariate almost stochastic dominance

397

Hence, the gain or loss in expected utility when switching from (Y1 , Y2 ) to (X 1 , X 2 ) can be written as E[u(X 1 , X 2 )] − E[u(Y1 , Y2 )]

b1  = u (1,0) (x1 , b2 ) Pr[Y1 ≤ x1 ] − Pr[X 1 ≤ x1 ] dx1 a1

b2 +

 u (0,1) (b1 , x2 ) Pr[Y2 ≤ x2 ] − Pr[X 2 ≤ x2 ] dx2

a2

+

b1 b2 

Pr[X 1 ≤ x1 , X 2 ≤ x2 ] − Pr[Y1 ≤ x1 , Y2 ≤ x2 ] u (1,1) (x1 , x2 )dx1 dx2 .

a1 a2

This expression can be traced back to Corollary 4 in Levy and Parouch (1974). The first two terms appearing in the expansion of E[u(X 1 , X 2 )] − E[u(Y1 , Y2 )] are clearly negative as u is non-decreasing and X i fsd Yi holds for i = 1, 2. Let us now show ε such that u (1,1) = 0 that the last one is also negative. To this end, consider u in Uca and denote γ = inf{−u (1,1) } > 0 δ = sup{−u (1,1) } > 0. Note that (4) ensures that  δ≤γ

γ 1 −1 ⇔ ≥ ε. ε γ +δ

(25)

Let S c denote the complement of S in [a1 , b1 ] × [a2 , b2 ]. Then,

b1 b2  a1 a2

=

Pr[X 1 ≤ x1 , X 2 ≤ x2 ] − Pr[Y1 ≤ x1 , Y2 ≤ x2 ] u (1,1) (x1 , x2 )dx1 dx2



+

Pr[Y1 ≤ x1 , Y2 ≤ x2 ]−Pr[X 1 ≤ x1 , X 2 ≤ x2 ]

S  c

≤δ

S 

+γ



Pr[Y1 ≤ x1 , Y2 ≤ x2 ]−Pr[X 1 ≤ x1 , X 2 ≤ x2 ]

 − u (1,1) (x1 , x2 ) dx1 d2



 − u (1,1) (x1 , x2 ) dx1 dx2

Pr[Y1 ≤ x1 , Y2 ≤ x2 ] − Pr[X 1 ≤ x1 , X 2 ≤ x2 ] dx1 dx2

S 

Pr[Y1 ≤ x1 , Y2 ≤ x2 ] − Pr[X 1 ≤ x1 , X 2 ≤ x2 ] dx1 dx2

Sc

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b1 b2     = −γ  Pr[X 1 ≤ x1 , X 2 ≤ x2 ] − Pr[Y1 ≤ x1 , Y2 ≤ x2 ]dx1 dx2 a1 a2

+ (γ + δ)



Pr[Y1 ≤ x1 , Y2 ≤ x2 ] − Pr[X 1 ≤ x1 , X 2 ≤ x2 ] dx1 dx2

S

≤ 0 by (25),

which ends the proof of the “⇒ ” part. Let us now turn to the “⇐” part. We assume that E[u(X 1 , X 2 )] ≤ E[u(Y1 , Y2 )] ε and we have to show that ( 5) holds. Let us proceed by contradiction holds for all u in Uca and assume that



Pr[Y1 ≤ x1 , Y2 ≤ x2 ] − Pr[X 1 ≤ x1 , X 2 ≤ x2 ] dx1 dx2

S

b1 b2     >ε  Pr[X 1 ≤ x1 , X 2 ≤ x2 ] − Pr[Y1 ≤ x1 , Y2 ≤ x2 ]dx1 dx2 .

(26)

a1 a2 ε such that Let us show that we can then construct a utility function u in Uca E[u(X 1 , X 2 )] > E[u(Y1 , Y2 )]. Let γ and δ be positive real numbers such that ε = γ (1,0) (x , b ) = 0, u (0,1) (b , x ) = 0, 1 2 1 2 γ +δ . Now consider a utility function u such that u (1,1) c (1,1) u = −γ on S and u = −δ on S, that is, a utility u proportional to (x1 − b1 )(x2 − b2 ) up to additive constants. We then see that

E[u(X 1 , X 2 )] − E[u(Y1 , Y2 )]

 =δ Pr[Y1 ≤ x1 , Y2 ≤ x2 ] − Pr[X 1 ≤ x1 , X 2 ≤ x2 ] dx1 dx2 S

+γ



Pr[Y1 ≤ x1 , Y2 ≤ x2 ] − Pr[X 1 ≤ x1 , X 2 ≤ x2 ] dx1 dx2

Sc

b1 b2 = −γ a1 a2

+(γ + δ)

     Pr[X 1 ≤ x1 , X 2 ≤ x2 ] − Pr[Y1 ≤ x1 , Y2 ≤ x2 ]dx1 dx2

 S

> 0, which ends the proof.

123

Pr[Y1 ≤ x1 , Y2 ≤ x2 ] − Pr[X 1 ≤ x1 , X 2 ≤ x2 ] dx1 dx2

Bivariate almost stochastic dominance

399

7.2 B Proof of Theorem 2 Let us start with the “⇒” part. By Corollary 1.6.12 in Denuit et al. (2005) applied to X i − ai and Yi − ai , we have E[u(X 1 , X 2 )] − E[u(Y1 , Y2 )]

b1  = u (1,0) (x1 , a2 ) Pr[X 1 > x1 ] − Pr[Y1 > x1 ] dx1 a1

b2 +

 u (0,1) (a1 , x2 ) Pr[X 2 > x2 ] − Pr[Y2 > x2 ] dx2

a2

+

b1 b2

Pr[X 1 > x1 , X 2 > x2 ]−Pr[Y1 > x1 , Y2 > x2 ] u (1,1) (x1 , x2 )dx1 dx2 . (27)

a1 a2

The first two terms appearing in the expansion of E[u(X 1 , X 2 )] − E[u(Y1 , Y2 )] are negative based on our assumptions that X i fsd Yi holds for i = 1, 2. Let us now show φ that the last one is also negative. To this end, consider u in Ucl such that u (1,1) = 0 and denote θ = inf{u (1,1) } > 0 θ¯ = sup{u (1,1) } > 0. Note that (4) ensures that θ¯ ≤ θ



1 θ −1 ⇔ ≥ φ. ¯θ + θ φ

(28)

Let Sˆ c denote the complement of Sˆ in [a1 , b1 ] × [a2 , b2 ]. Then,

b1 b2  a1 a2

=

Pr[X 1 > x1 , X 2 > x2 ] − Pr[Y1 > x1 , Y2 > x2 ] u (1,1) (x1 , x2 )dx1 dx2



Pr[X 1 > x1 , X 2 > x2 ] − Pr[Y1 > x1 , Y2 > x2 ] u (1,1) (x1 , x2 )dx1 dx2

ˆ

+

S  ˆc

≤ θ¯

S 

Pr[X 1 > x1 , X 2 > x2 ] − Pr[Y1 > x1 , Y2 > x2 ] u (1,1) (x1 , x2 )dx1 dx2 Pr[X 1 > x1 , X 2 > x2 ] − Pr[Y1 > x1 , Y2 > x2 ] dx1 d2

Sˆ

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+θ



Pr[X 1 > x1 , X 2 > x2 ] − Pr[Y1 > x1 , Y2 > x2 ] dx1 dx2

Sˆ c

b1 b2     = −θ  Pr[X 1 > x1 , X 2 > x2 ] − Pr[Y1 > x1 , Y2 > x2 ]dx1 dx2 a1 a2

+ (θ¯ + θ )



Pr[X 1 > x1 , X 2 > x2 ] − Pr[Y1 > x1 , Y2 > x2 ] dx1 dx2

Sˆ

≤ 0 by (25),

which ends the proof of the “⇒ ” part. With Eq. (27), the proof for the “⇐” part is similar to the one for “⇐” part in the proof for Theorem 1. Thus, it is omitted. 7.3 C Proof of Property 1 Let us first consider the implication with εca . Under (10) integration by parts gives E[X 1 X 2 ] − E[Y1 Y2 ]

b1 b2  Pr[X 1 ≤ x1 , X 2 ≤ x2 ] − Pr[Y1 ≤ x1 , Y2 ≤ x2 ] dx1 dx2 . = a1 a2

Now, 0≤



Pr[Y1 ≤ x1 , Y2 ≤ x2 ] − Pr[X 1 ≤ x1 , X 2 ≤ x2 ] dx1 dx2

S

⎛ ≤ ε⎝

 Sc

−



Pr[X 1 ≤ x1 , X 2 ≤ x2 ] − Pr[Y1 ≤ x1 , Y2 ≤ x2 ] dx1 dx2

⎞ Pr[X 1 ≤ x1 , X 2 ≤ x2 ] − Pr[Y1 ≤ x1 , Y2 ≤ x2 ] dx1 dx2 ⎠

S

so that 0≤



Pr[X 1 ≤ x1 , X 2 ≤ x2 ] − Pr[Y1 ≤ x1 , Y2 ≤ x2 ] dx1 dx2

Sc

1−ε + ε

123

 S

Pr[X 1 ≤ x1 , X 2 ≤ x2 ] − Pr[Y1 ≤ x1 , Y2 ≤ x2 ] dx1 dx2 .

Bivariate almost stochastic dominance

As ε < 0.5 ⇒ 1−ε ε > 1 and x2 ] dx1 dx2 ≤ 0 we have



0≤

  S

Pr[X 1 ≤ x1 , X 2 ≤ x2 ] − Pr[Y1 ≤ x1 , Y2 ≤

Pr[X 1 ≤ x1 , X 2 ≤ x2 ] − Pr[Y1 ≤ x1 , Y2 ≤ x2 ] dx1 dx2

Sc

+

401



Pr[X 1 ≤ x1 , X 2 ≤ x2 ] − Pr[Y1 ≤ x1 , Y2 ≤ x2 ] dx1 dx2

S

= E[X 1 X 2 ] − E[Y1 Y2 ]. This ends the proof since Cov[X 1 , X 2 ] − Cov[Y1 , Y2 ] = E[X 1 X 2 ] − E[Y1 Y2 ] under (10). φ Let us now turn to the implication with cl . Under (10), integration by parts gives E[X 1 X 2 ] − E[Y1 Y2 ]

b1 b2  Pr[X 1 > x1 , X 2 > x2 ] − Pr[Y1 > x1 , Y2 > x2 ] dx1 dx2 . = a1 a2

Now, 0≤



Pr[X 1 > x1 , X 2 > x2 ] − Pr[Y1 > x1 , Y2 > x2 ] dx1 dx2

Sˆ

⎛

⎜ ≤ φ ⎝−



Pr[X 1 > x1 , X 2 > x2 ] − Pr[Y1 > x1 , Y2 > x2 ] dx1 dx2

Sˆ c

+



⎞ ⎟ Pr[X 1 > x1 , X 2 > x2 ] − Pr[Y1 > x1 , Y2 > x2 ] dx1 d2 ⎠

Sˆ

so that 0≥



Pr[X 1 > x1 , X 2 > x2 ] − Pr[Y1 > x1 , Y2 > x2 ] dx1 dx2

Sˆ c

1−φ + φ



Pr[X 1 > x1 , X 2 > x2 ] − Pr[Y1 > x1 , Y2 > x2 ] dx1 dx2 .

Sˆ

123

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M. M. Denuit et al.

  As φ < 0.5 ⇒ 1−φ φ > 1 and Sˆ Pr[X 1 > x 1 , X 2 > x 2 ] − Pr[Y1 > x 1 , Y2 > x2 ] dx1 dx2 ≥ 0 we have

 Pr[X 1 > x1 , X 2 > x2 ] − Pr[Y1 > x1 , Y2 > x2 ] dx1 dx2 0≥ Sˆ c

+



Pr[X 1 > x1 , X 2 > x2 ] − Pr[Y1 > x1 , Y2 > x2 ] dx1 dx2

Sˆ

= E[X 1 X 2 ] − E[Y1 Y2 ]. This ends the proof. 7.4 D Proof of Property 2 ε . The announced result is valid if we can prove that the bivariate Consider v ∈ Ussd utility function u defined as

u(x1 , x2 ) = v(w1 x1 + w2 x2 ) ε . To this end, notice that belongs to Uca

u (1,1) (x1 , x2 ) = w1 w2 v (2) (w1 x1 + w2 x2 ) so that −u (1,1) (x1 , x2 ) = −w1 w2 v (2) (w1 x1 + w2 x2 )

 1 −1 ≤ w1 w2 inf{−v (2) (w1 x1 + w2 x2 )} ε  1 −1 . = inf{−u (1,1) } ε

This ends the proof. 7.5 E Proof of Theorem 3 Notice that

b Pr[X > t]dt = E[(X − x)+ ] x

where ξ+ denotes the positive part of the real ξ (equal to 0 if ξ is negative and to ξ otherwise). According to Corollary 1.6.10 in Denuit et al. (2005) applied to X − a and Y − a, we have

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403

E[v(X )] − E[v(Y )] 



b



= v (a) E[X ] − E[Y ] +

v a

(2)

(x)

b 

Pr[X > t] − Pr[Y > t] dtdx.

x

The first term is negative based on our assumptions. Let us now show that the second φ term of the above equation is also negative. Consider v in Url such that v (2) = 0 and denote λ = inf{v (2) } > 0 λ¯ = sup{v (2) } > 0. Note that (14) ensures that λ¯ ≤ λ



1 λ ≥ φ. −1 ⇔ φ λ¯ + λ

(29)

Then,

b v

(2)

(x)

a

b 

Pr[X > t] − Pr[Y > t] dtdx

x

=

v (2) (x)



b  x

+

v

(2)

(x)

b 

c

≤ λ¯

Pr[X > t] − Pr[Y > t] dtdx Pr[X > t] − Pr[Y > t] dtdx

x

b 

b  Pr[X > t]−Pr[Y > t] dtdx +λ Pr[X > t]−Pr[Y > t] dtdx

 x

c x



  = λ + λ¯

b



Pr[X > t] − Pr[Y > t] dtdx

 x

b  b     ¯ Pr[X > t] − Pr[Y > t] dt dx −λ  a

x

≤ 0 by (14), where c denotes the complement of  in [a, b]. It ends the proof of the “⇒” part. With Eq. (27), the “⇐ ” part is similar to the “⇐ ” part in the proof for Theorem 1. Thus, it is omitted.

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7.6 F Proof of Property 3 φ

Consider v ∈ Ucl . The announced result is valid if we can prove that the bivariate utility function u defined as u(x1 , x2 ) = v(w1 x1 + w2 x2 ) φ

belongs to Ucl . To this end, notice that u (1,1) (x1 , x2 ) = w1 w2 v (2) (w1 x1 + w2 x2 ) so that u (1,1) (x1 , x2 ) = w1 w2 v (2) (w1 x1 + w2 x2 )

 1 −1 ≤ w1 w2 inf{v (2) (w1 x1 + w2 x2 )} φ  1 −1 . = inf{u (1,1) } φ

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